Key Vocabulary: Wave Interference Standing Wave Node Antinode Harmonic Destructive Interference Constructive Interference 1. Work with two partners. Two will operate the Slinky and one will record the observations. Stretch the slinky to at least 5 m. While one end of the Slinky is held in a fixed position, send a pulse down the Slinky by quickly shaking one end.

a) What happens to the pulse when it reaches the far end of the Slinky?

2. Send a series of pulses down the Slinky by continuously moving one of its ends back and forth. Do not stop. Experiment with different frequencies until parts of the Slinky do not move at all. A wave whose parts appear to stand still is called a standing wave. 3. Set up the following standing waves:

A wave with one stationary point in the middle

A wave with two stationary points

A wave with as many stationary points as you can set up

4. You can simulate wave motion using a graphing calculator. Find a “window” that works to show your functions as follows:

Y1 = 4 sin x

Y2 = 4 sinx

Y3 = Y1 + Y2 a) Describe the waves you see on the screen.

b) Can you see that Y3 is equal to Y1 + Y2?

c) Find the amplitude of the crest of each function. How do they compare to each other?

5. Change your functions so that

Y1 = 4sin (x – 𝜋/4)

Y2 = 4 sin(x + 𝜋/4)

Y3 = Y1 + Y2 a) How many waves do you see on the screen? Compare the amplitude of the third wave to those

of the first two waves.

6. Edit again to:

Y1 = 4sin (x – 𝜋/2)

Y2 = 4 sin(x + 𝜋/2)

Y3 = Y1 + Y2 a) Describe the waves you see on the screen. Look for locations on the waves that always remain

zero. These locations are called nodes.

b) Draw the waves you see.

Label the nodes.

7. Edit again.

Y1 = 4sin (x – 3𝜋/4)

Y2 = 4 sin(x + 3𝜋/4)

Y3 = Y1 + Y2 a) Describe the waves you see on the screen.

b) Draw the waves you see.

Label the nodes.

c) Compare the amplitude of each wave. How does the amplitude of the third wave compare to

that of the first and the second wave?

8. Edit again.

Y1 = 4sin (x – 𝜋)

Y2 = 4 sin(x + 𝜋)

Y3 = Y1 + Y2 a) Describe the waves you see on the screen.

b) Draw the waves you see.

Locate and label the nodes.

c) What is the amplitude of the first wave?

What is the amplitude of the second wave?

How do they compare?

What Did You Learn? 1. What is a standing wave?

2. Describe in your own words how waves can “add.”

3. What properties must two pulses have if they are to cancel each other out when they meet on a

Slinky?

4. Make a standing wave. Draw it. Label its nodes and antinodes.

5. What is the distance, in wavelengths, between adjacent nodes in a standing wave pattern?

Explain your thinking.

Practical application: a) In photography, light can scatter off the camera lens. A thin coating is often placed on

the lens so that light reflecting off the front of the thin layer and light reflecting off the

lens will interfere with each other. How is this interaction helpful to the photographer?

b) Makers of noise reduction headsets say the devices “cancel” steady noises like airplane

engines, yet still allow the person wearing them to hear normal sounds such as voices.

How would these devices work? What principles of waves must be involved?

Standing Waves

Standing wave patterns are wave patterns produced in a medium when two waves of identical

frequencies (such as a slinky wave and its bounce-back reflection) interfere in such a manner to

produce points along the medium that always appear to be standing still. These points that have

the appearance of standing still are referred to as nodes. The points of maximum displacement

are called antinodes.

The simplest standing wave pattern that could be produced within a slinky is one that has nodes

at the two ends of the slinky and one antinode in the middle. The picture below shows the

vibrational pattern observed when the medium is seen vibrating in this manner.

The above pattern is called the first harmonic. Other wave patterns can be observed within the

slinky when it is vibrated at greater frequencies. For instance, if you vibrate the end with twice

the frequency as that associated with the first harmonic, then a second standing wave pattern can

be achieved. This standing wave pattern is characterized by nodes on the two ends of the slinky

and an additional node in the exact center of the slinky. As in all standing wave patterns, every

node is separated by an antinode. This pattern with three nodes and two antinodes is referred to

as the second harmonic and is depicted in the picture shown below.

If the slinky frequency is increased even more, then the third harmonic wave pattern can be

produced. The standing wave pattern for the third harmonic has an additional node and antinode

between the ends of the slinky. The pattern is depicted in the picture shown below.

c)

Observe that each consecutive harmonic is characterized by having one additional node and

antinode compared to the previous one. The table below summarizes the features of the standing

wave patterns for the first several harmonics.

HARMONIC # OF NODES # OF ANTINODES PATTERN

1st 2 1

2nd 3 2

3rd 4 3

4th 5 4

5th 6 5

6th 7 6

nth n + 1 n --

Questions:

1. What is the difference between a harmonic and the next level up?

2. What is a node?

3. What is an antinode?

4. What is a wave that has nodes and antinodes called?

What happens when two waves meet while they travel through the same medium?

What effect will the meeting of the waves have upon the appearance of the medium?

Will the two waves bounce off each other upon meeting (much like two billiard balls

would) or will the two waves pass through each other? These questions involving the

meeting of two or more waves along the same medium pertain to the topic of wave

interference.

What is Interference?

Wave interference is the phenomenon that occurs when two waves meet while

traveling along the same medium. The interference of waves causes the medium to

take on a shape that results from the net effect of the two individual waves upon the

particles of the medium. To begin our exploration of wave interference, consider two

pulses of the same amplitude traveling in different directions along the same medium.

Let's suppose that each displaced upward 1 unit at its crest and has the shape of a sine

wave. As the sine pulses move towards each other, there will eventually be a moment

in time when they are completely overlapped. At that moment, the resulting shape of

the medium would be an upward displaced sine pulse with an amplitude of 2 units. The

diagrams below depict the before and during interference snapshots of the medium for

two such pulses. The individual sine pulses are drawn in red and blue and the resulting

displacement of the medium is drawn in green.

Constructive Interference

This type of interference is sometimes called constructive interference. Constructive

interference is a type of interference that occurs at any location along the medium

where the two interfering waves have a displacement in the same direction. In this

case, both waves have an upward displacement; consequently, the medium has an

upward displacement that is greater than the displacement of the two interfering

pulses. Constructive interference is observed at any location where the two interfering

waves are displaced upward. But it is also observed when both interfering waves are

displaced downward. This is shown in the diagram below for two downward displaced

pulses.

In this case, a sine pulse with a maximum displacement of -1 unit (negative means a

downward displacement) interferes with a sine pulse with a maximum displacement of -

1 unit. These two pulses are drawn in red and blue. The resulting shape of the medium

is a sine pulse with a maximum displacement of -2 units.

Destructive Interference

Destructive interference is a type of interference that occurs at any location along

the medium where the two interfering waves have a displacement in the opposite

direction. For instance, when a sine pulse with a maximum displacement of +1 unit

meets a sine pulse with a maximum displacement of -1 unit, destructive interference

occurs. This is depicted in the diagram below.

In the diagram above, the interfering pulses have the same maximum displacement but

in opposite directions. The result is that the two pulses completely destroy each other

when they are completely overlapped. At the instant of complete overlap, there is no

resulting displacement of the particles of the medium. This "destruction" is not a

permanent condition. In fact, to say that the two waves destroy each other can be

partially misleading. When it is said that the two pulses destroy each other, what is

meant is that when overlapped, the effect of one of the pulses on the displacement of a

given particle of the medium is destroyed or canceled by the effect of the other pulse.

Recall from Lesson 1 that waves transport energy through a medium by means of each

individual particle pulling upon its nearest neighbor. When two pulses with opposite

displacements (i.e., one pulse displaced up and the other down) meet at a given

location, the upward pull of one pulse is balanced (canceled or destroyed) by the

downward pull of the other pulse. Once the two pulses pass through each other, there

is still an upward displaced pulse and a downward displaced pulse heading in the same

direction that they were heading before the interference. Destructive interference leads

to only a momentary condition in which the medium's displacement is less than the

displacement of the largest-amplitude wave.

The two interfering waves do not need to have equal amplitudes in opposite directions

for destructive interference to occur. For example, a pulse with a maximum

displacement of +1 unit could meet a pulse with a maximum displacement of -2 units.

The resulting displacement of the medium during complete overlap is -1 unit.

This is still destructive interference since the two interfering pulses have opposite

displacements. In this case, the destructive nature of the interference does not lead to

complete cancellation.

Interestingly, the meeting of two waves along a medium does not alter the individual

waves or even deviate them from their path. This only becomes an astounding behavior

when it is compared to what happens when two billiard balls meet or two football

players meet. Billiard balls might crash and bounce off each other and football players

might crash and come to a stop. Yet two waves will meet, produce a net resulting

shape of the medium, and then continue on doing what they were doing before the

interference.

The Principle of Superposition

The task of determining the shape of the resultant demands that the principle of

superposition is applied. The principle of superposition is sometimes stated as

follows:

When two waves interfere, the resulting displacement of the medium at any location is

the algebraic sum of the displacements of the individual waves at that same location.

In the cases above, the summing the individual displacements for locations of complete

overlap was made out to be an easy task - as easy as simple arithmetic:

Displacement of Pulse 1 Displacement of Pulse 2 = Resulting Displacement

+1 +1 = +2

-1 -1 = -2

+1 -1 = 0

+1 -2 = -1

Key vocabulary:

Boundary behaviour Medium

Fixed end reflection Free end reflection Inverted

Incident Pulse Reflected pulse Transmitted pulse

As a wave travels through a medium, it will often reach the end of the medium and

encounter an obstacle or perhaps another medium through which it could travel. For

example, a sound wave is known to reflect off canyon walls to produce an echo. A

sound wave traveling through air within a canyon reflects off the canyon wall and

returns to its original source. What affect does reflection have upon a wave? The

behavior of a wave (or pulse) upon reaching the end of a medium is referred to as

boundary behavior.

Fixed End Reflection

First consider an elastic rope stretched from end to

end. One end will be securely attached to a pole on a

lab bench while the other end will be held in the hand

in order to introduce pulses into the medium. Because

the right end of the rope is attached to a pole (which

is attached to a lab bench) (which is attached to the floor that is attached to the

building that is attached to the Earth), the last particle of the rope will be unable to

move when a disturbance reaches it. This end of the rope is referred to as a fixed end.

If a pulse is introduced at the left end of the rope, it will travel through the rope

towards the right end of the medium. This pulse is called the incident pulse since it is

incident towards (i.e., approaching) the boundary with the pole. When the incident

pulse reaches the boundary, two things occur:

A portion of the energy carried by the pulse is reflected and returns towards the left end of the rope. The disturbance that returns to the left after bouncing off the pole is known as the reflected pulse.

A portion of the energy carried by the pulse is transmitted to the pole, causing the pole to vibrate.

Because the vibrations of the pole are not visibly obvious, the energy transmitted to it is

not typically discussed. The focus of the discussion will be on the reflected pulse.

When one observes the reflected pulse off the fixed end, there are several notable

observations. First the reflected pulse is inverted. That is, if an upward displaced pulse

is incident towards a fixed end boundary, it will reflect and return as a downward

displaced pulse. Similarly, if a downward displaced pulse is incident towards a fixed end

boundary, it will reflect and return as an upward displaced pulse. See picture.

The inversion of the reflected pulse can be explained. When a crest reaches the end of

a medium ("medium A"), the last particle of the medium A receives an upward

displacement. This particle is attached to the first particle of the other medium

("medium B") on the other side of the boundary. As the last particle of medium A pulls

upwards on the first particle of medium B, the first particle of medium B pulls

downwards on the last particle of medium A. This is merely Newton's third law of

action-reaction. For every action, there is an equal and opposite reaction. The upward

pull on the first particle of medium B has little effect upon this particle due to the large

mass of the pole and the lab bench to which it is attached. The effect of the downward

pull on the last particle of medium A (a pull that is in turn transmitted to the other

particles) results in causing the upward displacement to become a downward

displacement. The upward displaced incident pulse thus returns as a downward

displaced reflected pulse. It is important to note that it is the heaviness of the pole and

the lab bench relative to the rope that causes the rope to become inverted upon

interacting with the wall. Other notable characteristics of the reflected pulse include:

The speed of the reflected pulse is the same as the speed of the incident pulse. The wavelength of the reflected pulse is the same as the wavelength of the incident

pulse. The amplitude of the reflected pulse is less than the amplitude of the incident pulse.

Of course, it is not surprising that the speed of the incident and reflected pulse are

identical since the two pulses are traveling in the same medium. Since the speed of a

wave (or pulse) is dependent upon the medium through which it travels, two pulses in

the same medium will have the same speed. This is also why the incident and reflected

pulses have the same wavelength. Every particle within the rope will have the same

frequency. Being connected to one another, they must vibrate at the same frequency.

Since the wavelength of a wave depends upon the frequency and the speed, two waves

having the same frequency and the same speed must also have the same wavelength.

Finally, the amplitude of the reflected pulse is less than the amplitude of the incident

pulse since some of the energy of the pulse was transmitted into the pole at the

boundary. The reflected pulse is carrying less energy away from the boundary

compared to the energy that the incident pulse carried towards the boundary. Since the

amplitude of a pulse is indicative of the energy carried by the pulse, the reflected pulse

has a smaller amplitude than the incident pulse.

Free End Reflection

Now consider what would happen if the end of the

rope were free to move. Instead of being securely

attached to a lab pole, suppose it is attached to a

ring that is loosely fit around the pole. Because the

right end of the rope is no longer secured to the

pole, the last particle of the rope will be able to

move when a disturbance reaches it. This end of the rope is referred to as a free end.

Once more if a pulse is introduced at the left end of the rope, it will travel through the

rope towards the right end of the medium. When the incident pulse reaches the end of

the medium, the last particle of the rope can no longer interact with the first particle of

the pole. Since the rope and pole are no longer attached and interconnected, they will

slide past each other. So when a crest reaches the end of the rope, the last particle of

the rope receives the same upward displacement; only now there is no adjoining

particle to pull downward upon the last particle of the rope to cause it to be inverted.

The result is that the reflected pulse is not inverted. When an upward displaced pulse is

incident upon a free end, it returns as an upward displaced pulse after reflection. And

when a downward displaced pulse is incident upon a free end, it returns as a downward

displaced pulse after reflection. Inversion is not observed in free end reflection.

Using the information above and the PhET “Wave on a String” simulation, complete the instructions below and answer the following questions.

1) Slide the bar labeled “Damping” all the way to the left (zero). By moving the wrench,

create a wave pulse.

2) Experiment with the wrench to see if you can get pulses with different amplitudes.

Now slide the damping bar to the right. What happens?

What do you think “damping” means?

In real life, do you think a string would behave more like the “zero damping” situation or

more like the damped string?

3) Put the damping level at zero again, and create another wave pulse. Notice what happens

to the wave pulse as it hits the fixed end. (This occurs because when the pulse exerts an

upward force on the clamp, the clamp exerts and equal and opposite (downward) force on

the string, reversing the orientation of the pulse.)

What do you predict would change if the clamp is removed?

Test your prediction by selecting “Loose end” and repeating the wave pulse. Was your

prediction correct?

Transmission of a Pulse Across a Boundary from Less to More Dense

Let's consider a thin rope attached to a thick rope, with each rope held at opposite ends

by people. And suppose that a pulse is introduced by the person holding the end of the

thin rope. If this is the case, there will be an incident pulse traveling in the less dense

medium (thin rope) towards the boundary with a more dense medium (thick rope).

Upon reaching the boundary, the usual two behaviors will occur.

A portion of the energy carried by the incident pulse is reflected and returns towards the left end of the thin rope. The disturbance that returns to the left after bouncing off the boundary is known as the reflected pulse.

A portion of the energy carried by the incident pulse is transmitted into the thick rope. The disturbance that continues moving to the right is known as the transmitted pulse.

The reflected pulse will be found to be inverted in situations such as this. During the

interaction between the two media at the boundary, the first particle of the more dense

medium overpowers the smaller mass of the last particle of the less dense medium.

This causes an upward displaced pulse to become a downward displaced pulse. The

more dense medium on the other hand was at rest prior to the interaction. The first

particle of this medium receives an upward pull when the incident pulse reaches the

boundary. Since the more dense medium was originally at rest, an upward pull can do

nothing but cause an upward displacement. For this reason, the transmitted pulse is not

inverted. In fact, transmitted pulses can never be inverted. Since the particles in this

medium are originally at rest, any change in their state of motion would be in the same

direction as the displacement of the particles of the incident pulse.

The Before and After snapshots of the two media are shown in the diagram below.

Comparisons can also be made between the characteristics of the transmitted pulse and

those of the reflected pulse. Once more there are several noteworthy characteristics.

The transmitted pulse (in the more dense medium) is traveling slower than the reflected pulse (in the less dense medium).

The transmitted pulse (in the more dense medium) has a smaller wavelength than the reflected pulse (in the less dense medium).

The speed and the wavelength of the reflected pulse are the same as the speed and the wavelength of the incident pulse.

How can these three characteristics be explained? First recall that the speed of a wave is dependent upon the properties of the medium. In this case, the transmitted and reflected pulses are traveling in two distinctly different media. Waves always travel fastest in the least dense medium. Thus, the reflected pulse will be traveling faster than the transmitted pulse. Second, particles in the more dense medium will be vibrating with the same frequency as particles in the less dense medium. Since the transmitted pulse was introduced into the more dense medium by the vibrations of particles in the less dense medium, they must be vibrating at the same frequency. So the reflected and transmitted pulses have the different speeds but the same frequency. Since the wavelength of a wave depends upon the frequency and the speed, the wave with the greatest speed must also have the greatest wavelength. Finally, the incident and the reflected pulse share the same medium. Since the two pulses are in the same medium, they will have the same speed. Since the reflected pulse was created by the vibrations of the incident pulse, they will have the same frequency. And two waves with the same speed and the same frequency must also have the same wavelength.

Transmission of a Pulse Across a Boundary from More to Less Dense

Finally, let's consider a thick rope attached to a thin rope, with the incident pulse

originating in the thick rope. If this is the case, there will be an incident pulse traveling

in the more dense medium (thick rope) towards the boundary with a less dense

medium (thin rope). There will be partial reflection and partial transmission at the

boundary. The reflected pulse in this situation will not be inverted. Similarly, the

transmitted pulse is not inverted (as is always the case). Since the incident pulse is in a

heavier medium, when it reaches the boundary, the first particle of the less dense

medium does not have enough mass to overpower the last particle of the more dense

medium. The result is that an upward displaced pulse incident towards the boundary

will reflect as an upward displaced pulse. For the same reasons, a downward displaced

pulse incident towards the boundary will reflect as a downward displaced pulse.

The Before and After snapshots of the two media are shown in the diagram below.

Comparisons between the characteristics of the transmitted pulse and the reflected

pulse lead to the following observations.

The transmitted pulse (in the less dense medium) is traveling faster than the reflected pulse (in the more dense medium).

The transmitted pulse (in the less dense medium) has a larger wavelength than the reflected pulse (in the more dense medium).

The speed and the wavelength of the reflected pulse are the same as the speed and the wavelength of the incident pulse.

The boundary behavior of waves in ropes can be summarized by the following:

The wave speed is always greatest in the least dense rope. The wavelength is always greatest in the least dense rope. The frequency of a wave is not altered by crossing a boundary. The reflected pulse becomes inverted when a wave in a less dense rope is heading

towards a boundary with a more dense rope. The amplitude of the incident pulse is always greater than the amplitude of the

reflected pulse.

Check Your Understanding

Case 1: A pulse in a more dense medium is traveling towards the boundary with a less

dense medium.

1. The reflected pulse in medium 1 ________ (will, will not) be inverted because

_______.

2. The speed of the transmitted pulse will be ___________ (greater than, less than, the

same as) the speed of the incident pulse.

3. The speed of the reflected pulse will be ______________ (greater than, less than,

the same as) the speed of the incident pulse.

4. The wavelength of the transmitted pulse will be ___________ (greater than, less

than, the same as) the wavelength of the incident pulse.

5. The frequency of the transmitted pulse will be ___________ (greater than, less than,

the same as) the frequency of the incident pulse.

Case 2: A pulse in a less dense medium is traveling towards the boundary with a more

dense medium.

6. The reflected pulse in medium 1 ________ (will, will not) be inverted because

_____________.

7. The speed of the transmitted pulse will be ___________ (greater than, less than, the

same as) the speed of the incident pulse.

8. The speed of the reflected pulse will be ______________ (greater than, less than,

the same as) the speed of the incident pulse.

9. The wavelength of the transmitted pulse will be ___________ (greater than, less

than, the same as) the wavelength of the incident pulse.

10. The frequency of the transmitted pulse will be ___________ (greater than, less

than, the same as) the frequency of the incident pulse.